If f(x)=4sinx3+cosxf(x)=4sinx3+cosx, then
f'(x)f′(x) =   

f'(3)f′(3) = 


We are given the function:

\[
f(x) = 4\sin(x^3) + \cos(x)
\]

### Step 1: Find \( f'(x) \)
To find the derivative \( f'(x) \), we will differentiate each term of the function.

1. **Derivative of \( 4\sin(x^3) \)**:
   - The outer function is \( \sin(u) \), where \( u = x^3 \).
   - The derivative of \( \sin(u) \) is \( \cos(u) \), and by the chain rule, we multiply by the derivative of \( u \) (which is \( 3x^2 \)).

\[
\frac{d}{dx}[4\sin(x^3)] = 4\cos(x^3) \cdot 3x^2 = 12x^2\cos(x^3)
\]

2. **Derivative of \( \cos(x) \)**:
   - The derivative of \( \cos(x) \) is \( -\sin(x) \).

\[
\frac{d}{dx}[\cos(x)] = -\sin(x)
\]

Now, putting these together:

\[
f'(x) = 12x^2 \cos(x^3) - \sin(x)
\]

### Step 2: Find \( f'(3) \

If f(x) = 4sin(x^3) + cos(x), then f'(x) = ? and f'(3) = ?

Find the derivative of the function f(x) = 4sin(x^3) + cos(x) using the chain rule and trigonometric derivatives. Detailed steps provided for differentiating each term and evaluating the derivative at x = 3. Suitable for students in advanced high school calculus or undergraduate studies.

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